Infinitely Many Surfaces with Prescribed Mean Curvature in the Presence of a Strictly Stable Minimal Surface

We construct infinitely many distinct hypersurfaces with prescribed mean curvature (PMC) for a large class of prescribing functions when $(M^{n+1},g)$ is a closed smooth manifold containing a minimal surface that is strictly stable (or more generally, admits a contracting neighborhood). In particular, we construct infinitely many distinct PMCs when $H_n(M,{\mathbb{Z}}_2)\mathrm{\ne }0$, or if $(M,g)$ does not satisfy the Frankel property. Our construction synthesizes ideas from Song’s construction of infinitely many minimal surfaces in the non-generic setting, Dey’s construction of multiple constant mean curvature surfaces, and Sun–Wang–Zhou’s min-max construction of free boundary PMCs.